Groups of order 12

Up to isomorphism there are five groups of order \(12\). The abelian ones are \[ \ZZ/4\ZZ\times\ZZ/3\ZZ\cong\ZZ/12\ZZ\qquad\text{and}\qquad(\ZZ/2\ZZ)^2\times\ZZ/3\ZZ, \] and the nonabelian ones are

1. the alternating group \(A_4\),

2. the dihedral group \(D_6\) of order \(12\),

3. the dicyclic group of order \(12\), \[ \gens{a,b\suchthat a^6=1,\ b^2=a^3,\ bab^{-1}=a^{-1}}\cong\ZZ/3\ZZ\rtimes\ZZ/4\ZZ, \] where a generator of \(\ZZ/4\ZZ\) acts on \(\ZZ/3\ZZ\) by inversion.